LMFDB - Newform orbit 435.2.q.d

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [435,2,Mod(41,435)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(435, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 0, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("435.41"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$N$	$=$	$435 = 3 \cdot 5 \cdot 29$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	435.q (of order $4$ , degree $2$ , minimal)

Newform invariants

comment:select newform

sage:traces = [36,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$3.47349248793$
Analytic rank:	$0$
Dimension:	$36$
Relative dimension:	$18$ over $\Q(i)$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

$q$ -expansion

The algebraic $q$ -expansion of this newform has not been computed, but we have computed the trace expansion.

\operatorname{Tr}(f)(q) =

36 q + 4 q^{2} + 2 q^{3} + 36 q^{5} - 8 q^{6} + 8 q^{7} - 4 q^{8} + 4 q^{10} + 12 q^{11} + 10 q^{12} - 28 q^{14} + 2 q^{15} - 60 q^{16} + 20 q^{17} + 32 q^{18} + 16 q^{19} - 12 q^{21} + 24 q^{24} + 36 q^{25}+ \cdots + 32 q^{99}+O(q^{100})

36 * q + 4 * q^2 + 2 * q^3 + 36 * q^5 - 8 * q^6 + 8 * q^7 - 4 * q^8 + 4 * q^10 + 12 * q^11 + 10 * q^12 - 28 * q^14 + 2 * q^15 - 60 * q^16 + 20 * q^17 + 32 * q^18 + 16 * q^19 - 12 * q^21 + 24 * q^24 + 36 * q^25 - 4 * q^26 - 22 * q^27 + 28 * q^29 - 8 * q^30 - 8 * q^31 + 16 * q^32 + 8 * q^33 + 8 * q^35 - 28 * q^36 - 4 * q^37 - 24 * q^38 - 24 * q^39 - 4 * q^40 - 48 * q^41 + 8 * q^42 + 4 * q^43 - 16 * q^44 + 20 * q^46 + 20 * q^47 - 66 * q^48 + 28 * q^49 + 4 * q^50 - 44 * q^52 - 24 * q^54 + 12 * q^55 + 84 * q^56 - 28 * q^57 - 64 * q^58 + 10 * q^60 + 20 * q^61 - 8 * q^62 - 32 * q^63 - 8 * q^66 - 60 * q^68 - 36 * q^69 - 28 * q^70 + 16 * q^71 + 64 * q^72 + 8 * q^73 + 2 * q^75 + 16 * q^76 - 32 * q^77 + 48 * q^78 + 12 * q^79 - 60 * q^80 - 60 * q^81 + 56 * q^82 + 100 * q^84 + 20 * q^85 - 8 * q^86 - 10 * q^87 - 24 * q^88 - 20 * q^89 + 32 * q^90 + 16 * q^92 - 24 * q^93 + 52 * q^94 + 16 * q^95 + 8 * q^96 + 4 * q^97 + 8 * q^98 + 32 * q^99

Embeddings

For each embedding $\iota_m$ of the coefficient field, the values $\iota_m(a_n)$ are shown below.

For more information on an embedded modular form you can click on its label.

comment:embeddings in the coefficient field

gp:mfembed(f)

Label	$a_{2}$			$a_{3}$					$a_{5}$	$a_{6}$			$a_{7}$	$a_{8}$			$a_{9}$			$a_{10}$
41.1	−1.93705	+	1.93705i	1.50032	−	0.865476i	−	5.50431i	1.00000	−1.22972	+	4.58265i	1.59227	6.78801	+	6.78801i	1.50190	−	2.59698i	−1.93705	+	1.93705i
41.2	−1.82725	+	1.82725i	−0.389527	+	1.68768i	−	4.67770i	1.00000	−2.37206	−	3.79558i	1.47752	4.89284	+	4.89284i	−2.69654	−	1.31480i	−1.82725	+	1.82725i
41.3	−1.34488	+	1.34488i	1.17986	−	1.26804i	−	1.61741i	1.00000	0.118585	+	3.29214i	−4.79407	−0.514537	−	0.514537i	−0.215844	−	2.99223i	−1.34488	+	1.34488i
41.4	−1.05880	+	1.05880i	1.69213	+	0.369716i	−	0.242102i	1.00000	−2.18308	+	1.40017i	1.34177	−1.86126	−	1.86126i	2.72662	+	1.25122i	−1.05880	+	1.05880i
41.5	−1.00351	+	1.00351i	−1.62379	+	0.602757i	−	0.0140454i	1.00000	1.02461	−	2.23435i	4.09600	−1.99292	−	1.99292i	2.27337	−	1.95750i	−1.00351	+	1.00351i
41.6	−0.780332	+	0.780332i	−0.476047	+	1.66535i		0.782165i	1.00000	−0.928048	−	1.67100i	0.394466	−2.17101	−	2.17101i	−2.54676	−	1.58557i	−0.780332	+	0.780332i
41.7	−0.602690	+	0.602690i	−1.20419	−	1.24496i		1.27353i	1.00000	1.47608	+	0.0245761i	0.173231	−1.97292	−	1.97292i	−0.0998697	+	2.99834i	−0.602690	+	0.602690i
41.8	−0.0236294	+	0.0236294i	1.03028	+	1.39231i		1.99888i	1.00000	−0.0572444	−	0.00855463i	3.42666	−0.0944913	−	0.0944913i	−0.877056	+	2.86893i	−0.0236294	+	0.0236294i
41.9	0.203088	−	0.203088i	0.407389	−	1.68346i		1.91751i	1.00000	−0.259155	−	0.424627i	4.71926	0.795601	+	0.795601i	−2.66807	−	1.37164i	0.203088	−	0.203088i
41.10	0.264256	−	0.264256i	1.60964	−	0.639569i		1.86034i	1.00000	0.256348	−	0.594368i	−1.21376	1.02012	+	1.02012i	2.18190	−	2.05896i	0.264256	−	0.264256i
41.11	0.398500	−	0.398500i	0.654884	+	1.60347i		1.68239i	1.00000	0.899957	+	0.378013i	−4.31678	1.46744	+	1.46744i	−2.14225	+	2.10018i	0.398500	−	0.398500i
41.12	0.602660	−	0.602660i	−1.60971	+	0.639395i		1.27360i	1.00000	−0.584771	+	1.35545i	−0.244261	1.97287	+	1.97287i	2.18235	−	2.05849i	0.602660	−	0.602660i
41.13	0.998333	−	0.998333i	−1.31362	−	1.12890i		0.00666265i	1.00000	−2.43844	+	0.184414i	2.30961	2.00332	+	2.00332i	0.451185	+	2.96588i	0.998333	−	0.998333i
41.14	1.39967	−	1.39967i	−1.39384	+	1.02821i	−	1.91814i	1.00000	−0.511758	+	3.39006i	−0.338767	0.114580	+	0.114580i	0.885569	−	2.86632i	1.39967	−	1.39967i
41.15	1.48187	−	1.48187i	0.154682	+	1.72513i	−	2.39186i	1.00000	2.78563	+	2.32719i	2.93783	−0.580678	−	0.580678i	−2.95215	+	0.533693i	1.48187	−	1.48187i
41.16	1.48848	−	1.48848i	1.06719	−	1.36422i	−	2.43117i	1.00000	−0.442117	−	3.61912i	−1.31013	−0.641785	−	0.641785i	−0.722191	−	2.91178i	1.48848	−	1.48848i
41.17	1.85345	−	1.85345i	−1.61221	−	0.633081i	−	4.87058i	1.00000	−4.16154	+	1.81476i	−4.54343	−5.32050	−	5.32050i	2.19842	+	2.04131i	1.85345	−	1.85345i
41.18	1.88783	−	1.88783i	1.32654	+	1.11368i	−	5.12777i	1.00000	4.60672	−	0.401834i	−1.70742	−5.90468	−	5.90468i	0.519415	+	2.95469i	1.88783	−	1.88783i
191.1	−1.93705	−	1.93705i	1.50032	+	0.865476i		5.50431i	1.00000	−1.22972	−	4.58265i	1.59227	6.78801	−	6.78801i	1.50190	+	2.59698i	−1.93705	−	1.93705i
191.2	−1.82725	−	1.82725i	−0.389527	−	1.68768i		4.67770i	1.00000	−2.37206	+	3.79558i	1.47752	4.89284	−	4.89284i	−2.69654	+	1.31480i	−1.82725	−	1.82725i

See all 36 embeddings

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
87.f	even	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	435.2.q.d	yes	36
3.b	odd	2	1		435.2.q.c	✓	36
29.c	odd	4	1		435.2.q.c	✓	36
87.f	even	4	1	inner	435.2.q.d	yes	36

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
435.2.q.c	✓	36	3.b	odd	2	1
435.2.q.c	✓	36	29.c	odd	4	1
435.2.q.d	yes	36	1.a	even	1	1	trivial
435.2.q.d	yes	36	87.f	even	4	1	inner

This newform subspace can be constructed as the kernel of the linear operator $T_{2}^{36} - 4 T_{2}^{35} + 8 T_{2}^{34} - 4 T_{2}^{33} + 125 T_{2}^{32} - 500 T_{2}^{31} + 1008 T_{2}^{30} + \cdots + 16$ T2^36 - 4*T2^35 + 8*T2^34 - 4*T2^33 + 125*T2^32 - 500*T2^31 + 1008*T2^30 - 484*T2^29 + 5142*T2^28 - 20516*T2^27 + 41864*T2^26 - 18892*T2^25 + 80983*T2^24 - 319664*T2^23 + 665352*T2^22 - 265528*T2^21 + 558414*T2^20 - 2119484*T2^19 + 4437040*T2^18 - 1478868*T2^17 + 1926099*T2^16 - 6475020*T2^15 + 12321160*T2^14 - 4028452*T2^13 + 3356554*T2^12 - 8457516*T2^11 + 12041536*T2^10 - 5557036*T2^9 + 2569441*T2^8 - 3164032*T2^7 + 3541120*T2^6 - 2007280*T2^5 + 664185*T2^4 - 114240*T2^3 + 8192*T2^2 + 512*T2 + 16 acting on $S_{2}^{\mathrm{new}}(435, [\chi])$ .

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Embeddings:		e.g. 1-3 or 41.18
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